daubechies wavelet

Named after Ingrid Daubechies, the Daubechies wavelet is a wavelet used to convolve image data. The wavelets can be orthogonal, when the scaling functions have the same number of coefficients as the wavelet functions, or biorthogonal, when the number of coefficients differ. The JPEG 2000 compression standard uses the biorthogonal Daubechies 5/3 wavelet (also called the LeGall 5/3 wavelet) for lossless compression and the Daubechies 9/7 (also known as the Cohen-Daubechies-Fauraue 9/7 or the "CDF 9/7") for lossy compression. In general Daubechies wavelet has extremal phase and highest number of vanishing moments for defined support width. The wavelet is also easy to put into practice with minimum-phase filters. Daubechies wavelet is widely used in solving a broad range of problems, e.g. self-likely properties of a signal or fractal problem, signal discontinuities, etc.

Orthogonal wavelets

Daubechies orthogonal wavelets D2-D20 (even index numbers only) are commonly used. The index number refers to the number of coefficients. Each wavelet has a number of zero moments or vanishing moments equal to half the number of coefficients. For example D2 has one vanishing moments, D4 has two moments, etc. A vanishing moment refers to the wavelets ability to represent polynomial behaviour or information in a signal. For example, D2, with one moment, easily encodes polynomials of one coefficient, ie. constant signal components. D4 encodes polynomials of two coefficients, ie constant and linear signal components, D6 encodes 3-polynomials, ie constant, linear and quadratic signal components.

Orthogonal wavelet coefficients

Both the Scaling Function (Low-Pass Filter) and the Wavelet Function (High-Pass Filter) must be normalised by a factor

\frac{1}{\sqrt{2}}

. Below are the coefficients for the scaling functions for D2-20. The wavelet coefficients are derived by reversing the order of the scaling function coefficients and then reversing the sign of every second one. (ie. D4 wavelet = {-0.1830127, -0.3169873, 1.1830127, -0.6830127}) Mathematically, this looks like

b^k = (-1)^{k} C_{N - 1 - k}

where k is the coefficient index, b is a wavelet coefficient and c a scaling function coefficient. N is the wavelet index, ie 2 for D2. Orthogonal Daubechies coefficients>

2 (Haar)	D4	D6	D8	D10	D12	D14	D16	D18	D20
\|0.6830127 \|0.47046721 \|0.32580343 \|0.22641898 \|0.15774243 \|0.11009943 \|0.07695562 \|0.05385035 \|0.03771716
\|1.1830127 \|1.14111692 \|1.01094572 \|0.85394354 \|0.69950381 \|0.56079128 \|0.44246725 \|0.34483430 \|0.26612218
\|0.3169873 \|0.650365 \|0.8922014 \|1.02432694 \|1.06226376 \|1.03114849 \|0.95548615 \|0.8553430 \|0.74557507
\| -0.1830127 \| -0.19093442 \| -0.03967503 \|0.19576696 \|0.44583132 \|0.66437248 \|0.82781653 \|0.92954571 \|0.97362811
\| \| -0.12083221 \| -0.26450717 \| -0.34265671 \| -0.31998660 \| -0.20351382 \| -0.02238574 \|0.18836955 \|0.39763774
\| \|0.0498175 \|0.0436163 \| -0.04560113 \| -0.18351806 \| -0.31683501 \| -0.40165863 \| -0.41475176 \| -0.35333620
\| \| \|0.0465036 \|0.10970265 \|0.13788809 \|0.1008467 \|6.68194092e-4 \| -0.13695355 \| -0.27710988
\| \| \| -0.01498699 \| -0.00882680 \|0.03892321 \|0.11400345 \|0.18207636 \|0.21006834 \|018012745
\| \| \| \| -0.01779187 \| -0.04466375 \| -0.05378245 \| -0.02456390 \|0.04345268 \|0.13160299
\| \| \| \|4.71742793e-3 \|7.83251152e-4 \| -0.02343994 \| -0.06235021 \| -0.09564726 \| -0.10096657
\| \| \| \| \|6.75606236e-3 \|0.01774979 \|0.01977216 \|3.54892813e-4 \| -0.04165925
\| \| \| \| \| -1.52353381e-3 \|6.07514995e-4 \|0.01236884 \|0.03162417 \|0.04696981
\| \| \| \| \| \| -2.54790472e-3 \| -6.88771926e-3 \| -6.67962023e-4 \|5.10043697e-3
\| \| \| \| \| \| 5.00226853e-4 \| -5.54004549e-4 \| -6.05496058e-3 \| -0.01517900
\| \| \| \| \| \| \|9.55229711e-4 \|2.61296728e-3 \|1.97332536e-3
\| \| \| \| \| \| \| -1.66137261e-4 \|3.25814671e-4 \|2.81768659e-3
\| \| \| \| \| \| \| \| -3.56329759e-4 \| -9.69947840e-4
\| \| \| \| \| \| \| \| -5.5645514e-5 \| -1.64709006e-4
\| \| \| \| \| \| \| \| \|1.32354367e-4
\| \| \| \| \| \| \| \| \| -1.875841e-5

Biorthogonal wavelets

9/7 Coefficients>

!Low Pass !High Pass
\|0.6029490182363579 \|1.115087052456994
+/-1	0.2668641184428723	-0.5912717631142470
+/-2	-0.07822326652898785	-0.05754352622849957
+/-3	-0.01686411844287495	0.09127176311424948
+/-4	0.026748741080976
\| \|

External links

Hardware implementation of wavelets

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